Strategy parameter adaptation in evolution strategies

ABSTRACT

The present invention relates to a optimization method based on an evolution strategy according to which a model/structure/shape/design to be optimized is described by parameter sets comprising object parameters. The object parameters are mutated to create offsprings of the parameter set. The quality of the offsprings is evaluated. The parameter set furthermore comprises at least one strategy parameter representing the step-size of the mutation (f.e. the variance of the normal distribution) of associated object parameters. The number of object parameters as well as the number of associated strategy parameters can be adapted during the optimization process. The value of newly inserted strategy parameters can be estimated based on the information of correlated object parameters.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is related to and claims priority from European PatentApplication No. 01 104 723.0 filed on Feb. 26, 2001.

FIELD OF THE INVENTION

The present invention relates to an optimization method based on anevolution strategy, to a method for optimizing spline coded structureson the basis of an evolution strategy, to a computer software forexecuting such a method as well as to the use of such a method for theoptimization of the shape of aerodynamic or hydrodynamic structures.

BACKGROUND OF THE INVENTION

In the field of evolution strategy, basic principles of naturalevolution are used for generating or optimizing technical structures.Basic operations are mutation and recombination as a method formodifying structures or parameters. To eliminate unfavorablemodifications and to proceed with modifications which increase theoverall quality of the system, a selection operation is used. Principlesof the evolution strategy can be found for example in Rechenberg, Ingo(1994) “Evolutions strategie”, Friedrich Frommann Holzboog Verlag.

With reference to FIG. 1 at first the known cycle of an evolutionstrategy will be explained.

In a step 1 the object parameters to be optimized are encoded as realnumbers in a vector called individual or chromosome. One individual canalternatively also consist of several vectors or chromosomes. A numberof such individuals are generated that comprise the initial parentgeneration and the quality (fitness) of each individual in the parentgeneration is evaluated. In a step S2 the parents are reproduced byapplying operators called mutation and recombination. Thus, a newgeneration is produced in step S3, which is called the offspringgeneration. The quality of the offspring individuals is evaluated usinga fitness function which is the objective of the optimization in stepS4. Finally, depending on the calculated quality value, step S5 selects(possibly stochastically) the best offspring individuals (survival ofthe fittest) which are used as parents for the next generation cycle ifthe termination condition in step S6 is not satisfied.

In particular for this application the real number vector, the objectparameter, represents the coding for a spline which describes a two orhigher dimensional body. This mapping from the parameter vector to thespline encoded structure is usually referred to as the genotype (=theparameter vector)−phenotype (=the two or higher dimensional body)mapping. In order to determine the fitness in step S4, first thegenotype is mapped to the phenotype and then the quality of thephenotype is derived. A particular example is the parameterization of anairfoil (=the phenotype) by a real-valued vector (=genotype) describinga spline which determines the geometry of the airfoil. Finally, thequality of the airfoil can be determined, e.g. by methods known fromcomputational fluid dynamics.

The mutation operator for the reproduction in step S2 is realized byadding normal or Gaussian distributed random numbers to the alreadyexisting elements of the parameter set, the so-called chromosome. Thestep size of the mutation can be controlled by modifying the variance ofthe normal or Gaussian probability density function. The variances arecalled strategy parameters. In higher dimensions the strategy parameterscan consist of all or some entries of the covariance matrix of thehigher dimensional Gaussian probability density function. This method tocontrol the mutation by means of strategy parameters particularlydistinguishes evolution strategy from other evolutionary algorithms likegenetic algorithms (GA) or genetic programming (GP).

Appropriate values for the strategy parameters are important for theconvergence of the algorithm. The appropriate values depend on theproblem and the actual position in the solution space. To adapt themutation width online, different adaptation strategies are used. Somesimple methods are the ⅕ rule (see Rechenberg) or the so-called mutativestep size control. More complex methods are, for example, thede-randomized adaptation method or the co-variance matrix adaptation.

Evolution strategies have been shown to outperform other evolutionaryalgorithms like genetic algorithms or genetic programming forreal-valued parameter optimization problems due to the above-mentionedpossibilities of the adaptation or self-adaptation of the step size(s)of the mutation operator.

BRIEF SUMMARY OF THE INVENTION

A known problem for the application of evolution strategy to designoptimization is that the number of parameters which describe the model(the so-called object parameters) is fixed in evolution strategy.Therefore, an automatic refinement of the phenotype represented by theobject parameters during the optimization is impossible.

Mathematically, the mutation operation step can be expressed by thefollowing equation (1):{right arrow over (x)} ^(O) ={right arrow over (x)} ^(P) +N({right arrowover (0)},{right arrow over (σ)}_(P))  (1)thereby {right arrow over (x)}^(O) represents the parameter set of theoffspring, {right arrow over (x)}^(P) represents the parameter set ofthe parents and N({right arrow over (0)},{right arrow over (σ)}_(P))represents a vector whose elements are normal distributed random numberswith zero mean with variances σ_(P,1) ². As can be seen from FIG. 3, alarger strategy parameter σ_(P,1) increases the probability for largersteps in the offspring mutation.

More generally, the mutation operation can be represented by thefollowing equation (2):{right arrow over (x)} ^(O) ={right arrow over (x)} ^(P)+{right arrowover (δ)}  (2)wherein

$\begin{matrix}{\overset{arrow}{\delta} \sim {\frac{\sqrt{\det( \sum^{- 1} )}}{( {2\;\pi} )^{n/2}}{\exp( {{- \frac{1}{2}}( {\overset{arrow}{x} - \overset{arrow}{u}} )^{T}{\sum^{- 1}( {\overset{arrow}{x} - \overset{arrow}{u}} )}} )}}} & (3)\end{matrix}$

FIG. 3 a and b show the resulting probability distribution for placingan offspring, wherein FIG. 3 a shows a 3D-plot and 3 b shows acontour-plot. δ is a random vector with a general Gaussian probabilitydensity function as specified in Eq. (3), if the covariance matrix Σ isdiagonal, Eq. (2) is reduced to Eq. (1) where the elements of the vector({right arrow over (σ)}_(P)), the variances σ_(P,1) ², are the diagonalelements of Σ.

The covariance matrix Σ has to be adapted to the local quality function.FIG. 4 d shows the self-adaptation of the strategy parameters(covariance matrix Σ) to the local quality function. Particularly, FIG.3 c shows the result of four cycles of the reproduction process, whereinthe 3 d surface represents the quality function. By the self-adaptationof the strategy parameters to the local quality function regarding thedirection and the step size a good convergence towards the absolutemaximum of the local quality function can be achieved.

It is the object of the present invention to deal with the problem ofvariable parameter set length in evolution strategy in particular incontext with the self-adaptation property of the strategy parameter set.With other words, the present invention proposes a technique forautomatically refining the phenotype described by the parameter setduring the optimization.

This object is achieved by means of the features of the independentclaims. The dependent claims develop further the central idea of thepresent invention.

According to the present invention, the size of the describing parameterset during the evolution process can be adapted. For example, in thebeginning a very small parameter set can be used to find a roughapproximation of the optimal shape, which can be achieved very fast.During the optimization, the parameter set can be expanded to describemore and more complex shapes. However, the parameter set can not only bechanged by inserting new object parameters and appropriate strategyparameters, but also by removing some parameters. This can be the casef.e. if an object parameter shows to be not sufficiently useful forfurther optimization.

When expanding the number of object parameters in the parameter sets,the associated number of strategy parameters has also to be expanded byinserting new strategy parameters. The present invention thereforeproposes a technique for estimating the value of newly inserted strategyparameters.

According to a first aspect of the present invention therefore anoptimization method based on an evolution strategy is proposed. A model,structure, shape or design to be optimized is thereby described by aparameter set comprising object parameters which are mutated and/orrecombined to create offsprings of the parameter set. The quality of theoffspring is then evaluated. The parameter set furthermore comprises atleast one strategy parameter representing the variance of the mutationof associated object parameters. Thereby, the number of objectparameters as well as the number of associated strategy parameters canbe adapted during the optimization process.

For example the object parameters and strategy parameters can beselectively inserted and/or removed.

The value of a newly inserted strategy parameter can be estimated basedon the information of strategy parameters of correlated objectparameters.

The position and/or the time of a removal or an insertion of an objectparameter and an associated strategy parameter can be determined by arandom function.

The changes of the object parameter set and the associated strategyparameters can be performed such that it has no influence on the resultof the evaluation step, thus the change of the parameters can beselectively neutral.

According to a further aspect of the present invention, an optimizationmethod based on an evolution strategy is proposed. Thereby the structureof a parameter set defined by the number and/or position of the objectparameters and strategy parameters can be mutated to create theseoffsprings.

According to a still further aspect of the present invention, a methodfor the optimization of spline coded problems is proposed. The objectparameters in this case can comprise control and knot points, whereinnew control points and strategy parameters can be inserted.

The values of newly inserted strategy parameters can be estimated on thebasis of the values of the strategy parameters of neighboring controlpoints.

According to a further aspect of the present invention, a method foroptimizing spline coded shapes on the basis of an evolution strategy isproposed. Thereby a model, structure, shape or design to be optimized isdescribed by a parameter set comprising object parameters representingcontrol points and knot points of the spline coded shapes and at leastone strategy parameter representing the variance of the mutation ofassociated object parameters. The object parameters and strategyparameters are mutated to create offsprings of the parameter set.Finally, the quality of the offsprings is evaluated. According to theinvention, the step of mutating comprises the step of determination of acontrol point insertion position, insertion of the control point,adjustment of the knot vector, insertion of the appropriate strategyparameter for the inserted control point, weighted averaging of thestrategy parameter values of the modified control points, and assigningthe weighted average value as the value of the inserted strategyparameter.

According to a further aspect of the present invention, a computersoftware program for executing a method as said above is proposed.

Finally, the invention proposes the use of such a method for theoptimization of the shape of aerodynamic or hydrodynamic structures.

Further features, objects and advantages of the present invention willbecome evident for the man skilled in the art when reading the followingdescription of embodiments of the present invention when taken inconjunction with the figures of the enclosed drawings.

DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic representation of an evolutionary algorithm.

FIG. 2 is a graphic representation showing the influence of the strategyparameter value.

FIG. 3 shows the relation between the strategy parameter and thedirection and step size of the evolution strategy process.

FIG. 4 shows schematically the structure mutation for individual stepsize adaptation of the strategy parameter. Note, that the neighboringcontrol points and strategy parameters might also be affected by thestructure mutation.

FIG. 5 shows schematically the structure mutation for the covariancematrix adaptation of the strategy parameter. Note, that the neighboringrows and columns in the covariance matrix might also be affected.

FIG. 6 shows a flow-chart for the process of adapting the structure of aparameter set by inserting/removing strategy and object parameters.

FIG. 7 shows different hierarchies of mutations.

FIG. 8 is a schematic flow chart showing the mutation of the structureof a parameter set.

FIGS. 9 and 10 show an application example of the present invention: theoptimization of spline coded shapes.

FIG. 11 shows the application of the present invention to theoptimization of the shape of an aerodynamic or hydrodynamic body.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The principles of evolution strategy have already been explained withreference to FIGS. 1 to 4. The general difference between the presentinvention and the prior art in evolution strategy is that the structureof the genotype, i.e., the structure of the parameter set including thestrategy parameters, can also be mutated. With other words, both thenumber of object parameters and strategy parameters of a parameter setcan be changed. Therefore, the optimization process can start with arelatively low number of parameters and the complexity of the parameterset can be refined during the optimization process.

With reference to FIGS. 4 and 5, the time point and/or the location forthe insertion of a new object parameter in the existing parameter setcan be defined for example by means of a random function. When insertinga new object parameter, at the corresponding location of the strategyparameters a new strategy parameter is inserted such that the old“chromosome” (parameter set comprising object parameters and strategyparameters) is extended to a new chromosome. The concept shown in FIG. 4refers to the so called individual mutative step size adaptation. Note,that neighboring object and strategy parameters might also change afterthe modification.

The structure mutation of the covariance matrix is schematically shownin FIG. 5. The strategy parameter according to this embodiment isrepresented as the covariance matrix Σ (see equation (3) further above).When inserting a new object parameter in the old chromosome to create anextended chromosome, the covariance matrix Σ is adapted by inserting anew row and a new column, thus generating a new expanded covariancematrix Σ′. The parameter set according to this representation iscomposed of the object parameter set and the strategy parameter, e.g.the co-variance matrix Σ. Note, that neighboring rows and columns of thecovariance matrix might be affected by the modification.

The present invention is for example applied to the field ofoptimization of spline coded problems. An example of a spline codedproblem is shown in FIGS. 10 to 12. Typically in such spline codedproblems, all attributes of the shape are encoded in the parameter set(control points, knot points which comprise the object parameters andthe strategy parameters). Attributes which are not encoded cannot berepresented and therefore cannot be subject of the optimization. As aconsequence thereof, attributes cannot evolve during the optimization aslong as they are not included in the parametric description of theparameter set.

The present invention now allows for an online modification of theparametric description (modification of the structure of the genotype)in order to allow new attributes in the encoding and therefore also inthe shape (=the phenotype) to be inserted. Note, that the parameter setcan not only be expanded by inserting new object and strategyparameters, but also be reduced by removing these parameters. This canbe the case f.e. if an object parameter shows to be not sufficientlyuseful for further optimization.

As has already been said, one of the main advantages of the evolutionstrategies compared to other evolutionary algorithms is the use of thestrategy parameters which can represent a preferred direction and stepsize for a further search. The strategy parameters are also adapted(mutated) in evolution strategies during the optimization to the localproperties of the quality function. When new components (new controlpoints in the case of spline coded problems) are inserted in theencoding or the encoding is modified, the values of the newly insertedstrategy parameters have to be calculated.

According to the present invention, when determining a value of a newlyinserted strategy parameter, the information of correlated components,e.g. the information of correlation parameters associated withcorrelated object parameters is used.

In the case of spline encoded structures, strategy parameters of newpoints are usually similar to strategy parameters of their neighboringpoints. Therefore, in the case of spline encoded structures, thestrategy parameter value can be estimated from the strategy parametervalues of neighboring control points.

The optimization process for a spline encoded problem or structure cantherefore be represented in a simple way as follows:

-   -   definition of an initial spline encoding (object parameter set)        for the geometric shape (for example hydrodynamic or aerodynamic        bodies such as an airfoil) and of an initial strategy parameter        set (f.e. covariance matrxi)    -   performing the steps of a standard evolution strategy for        optimization    -   introducing (possibly at random) new control point(s) in the        spline encoding (new parameters) due to any predefined measure,        f.e. stagnation of improvement with current encoding or as an        additional mutation operator    -   estimation of the new strategy parameter values based on        existing strategy parameters, and    -   continuation of the standard evolution strategy for        optimization.

This process usable for example for spline encoded problems will now beexplained with reference to FIG. 6.

In step S1 the initial encoding of the problem, for example representedby control and knot points, is set. In step S2 it is decided whether theencoding should be mutated. This decision can for example be triggeredby a random function or a function that depends on the progress of theoptimization. In case the result of step S2 is yes, in step S3 it isdecided whether a new control point is to be inserted. If the answer ofthe decision step S3 is yes (which can be again decided on the basis ofa random function or some kind of sensitivity analysis of the splinecoding), the position for the newly to be inserted control point has tobe found in step S4. Therefore the index i of the newly to be insertedpoint representing the position of the newly inserted point in theparameter set has to be found in step S4 and the new control point isinserted at the index i in step S5. Finally in a step S6, the value ofthe correspondingly to be inserted strategy parameters strat_(i) iscalculated. The value of the newly inserted strategy parametersstrat_(i) is a function ƒ(strat _(i−1),strat _(i+1)) of the values ofthe neighboring strategy parameters i−1 and i+1 or in more general termsa function of a subset of all old strategy parameters.

In case it is decided in step S3 to mutate the coding without insertionof a new control point, the removal position has to be chosen in step S7and the corresponding control point is removed in step S8.

Finally, the parameter set including the control points, knot points andstrategy parameter is mutated in step S9. The resulting structure ofthis mutation step S9 is then evaluated in step S10 using a so-calledfitness function, f.e. measuring the fluid-dynamics properties of thespline coded shape.

In step S11, it is checked whether all individuals, i.e., all parametersets (comprised of the object parameters and the strategy parameters)have been mutated. If this is not the case, the process goes back tostep S2.

If the decision in S11 is positive, the n best individuals out of the mindividuals generated in step S9 and evaluated in step S10 are selectedin step S12 (“survival of the fittest”); this corresponds to a (n,m)selection, other selection methods like (n+M) or tournament selection,etc. can be applied. After the selection process, the optimization iseither stopped, S13, in case some pre-defined criterion has been reachedor a new generation is started with the reproduction step in S14 and anew mutation step in S1.

FIG. 7 shows the different hierarchies of mutations possible accordingto the present invention.

The mutation of the parameter set (object parameters S15 and strategyparameters S16) in a step S9 is already known. However, according to thepresent invention, also the structure, e.g. the number and position ofthe object parameters and strategy parameters of the parameter set canbe mutated in a step S17 (comprising the steps S2 to S8 in FIG. 6).

FIG. 8 summarizes the optimization processing as shown in FIG. 6. Thegenotype is first extended by inserting a new object parameter in stepS3 a. The value of this new parameter is estimated from the existingobject parameter values in step S3 b, f.e. in order to minimize theinitial variation of the phenotype following the extension of thegenotype. Once the position and the value of the new object parameterhas been calculated in steps S3 a and S3 b, the position and the valueof the new strategy parameter is calculated using already presentstrategy parameters corresponding to the object parameters used in theestimation of the value for the new object parameter (step S3 b).

Finally, an extended genotype with optimized strategy parameters for thenew object parameters is generated.

A simple example for the application of the present invention on aspline encoded structure is shown in FIGS. 9 and 10. The problem in thiscase is the approximation of a circle by a polygon with a fixed (FIG. 9)and a variable (FIG. 10) number of edges. The present invention (FIG.10) has the advantage that this simple optimization process can bestarted with a structure with few parameters, f.e. a triangle, whosenumber is subsequently increased during the optimization. Due to theiterative development the evolution does not get stuck like in FIG. 9,where the degree of freedom is too large for the strategy parameters toadapt.

The mutation process for such spline encoded structures can besummarized as follows:

-   -   1. Based on a simple structure (small number of object        parameters), a control point insertion position is determined.    -   2. The control point is inserted at the determined insert        position.    -   3. The knot vector is changed according to the inserted control        point.    -   4. To calculate the value of the new strategy parameter for the        new control point, a weighted averaging of the strategy        parameters of the neighboring control points is performed, and    -   5. The weighted average value is assigned as value for the new        inserted strategy parameter.

According to the description, the present invention finds applicationfor all structure encodings in which the optimal strategy parameter canbe estimated from already existing strategy parameters.

FIG. 11 visualizes the estimation of strategy parameters and theapplication of the present invention to the design of airfoils. R5thereby designates the airfoil. R6 designates the control polygon forthe spline which approximates the airfoil. R9 designates an alreadypresent control point and the ellipsoid R10 around this control point R9visualizes the probability density distribution according to thestrategy parameter which is the expected area for modification. R7designates a new inserted control point having a strategy parameter R8represented by the gray-shaded area.

Further examples for the application of the present invention apart fromairfoil shapes are compressor/turbine blades for gas turbines (statorand rotor blades) and other aerodynamic or hydrodynamic structures.Other fields of application are architecture and civil engineering,computer science, wing design, engine valve design and schedulingproblems.

1. A computer based method of optimizing one of a model, structure,shape or design representing an aerodynamic structure or a hydrodynamicstructure based on an evolution strategy, comprising: describing one ofthe model, structure, shape or design representing an aerodynamicstructure or a hydrodynamic structure to be optimized using a parameterset comprising a first number of object parameters; creating offspringsof the parameter set by modifying the object parameters, wherein saidmodifying includes at least one of mutating the object parameters andrecombining the object parameters; evaluating quality of the offsprings;wherein the parameter set comprises at least one strategy parameterrepresenting a step-size of the mutation of associated objectparameters; modifying said first number of the object parameters to asecond number of object parameters and modifying a first number ofassociated strategy parameters to a second number of associated strategyparameters during optimization to optimize one of said model, structure,shape or design; and storing one of said optimized model, optimizedstructure, optimized shape or optimized design in a computer storage. 2.The optimization method of claim 1 further comprising altering theobject parameters and the strategy parameters, wherein said alteringincludes at least one of selectively inserting or removing an objectparameter and a strategy parameter.
 3. The optimization method of claim2, further comprising estimating a value of a newly inserted strategyparameter based on information of strategy parameters associated withcorrelated object parameters.
 4. The optimization method of claim 1,further comprising estimating a value of a newly inserted strategyparameter based on information of strategy parameters associated withcorrelated object parameters.
 5. The optimization method of claim 1,further comprising determining a position of altering an objectparameter and an associated strategy parameter using a random function.6. The optimization method of claim 5, further comprising determining atime of altering the object parameter and the associated strategyparameter using a random function.
 7. The optimization method of claim1, further comprising determining a time of altering an object parameterand an associated strategy parameter using a random function.
 8. Theoptimization method of claim 1, further comprising determining aposition of altering an object parameter and an associated strategyparameter by progress of the evolutionary optimization.
 9. Theoptimization method of claim 8, further comprising determining a time ofsaid altering of said object parameter and the associated strategyparameter by the progress of the evolutionary optimization.
 10. Theoptimization method of claim 8, further comprising determining a time ofsaid altering of said object parameter and the associated strategyparameter by the progress of the evolutionary optimization.
 11. Theoptimization method of claim 1, wherein the mutating of the objectparameters does not directly influence the result of the evaluatingstep.
 12. A computer based method of optimizing one of a model,structure, shape or design representing an aerodynamic structure or ahydrodynamic structure based on an evolution strategy, comprising:describing one of the model, structure, shape or design representing anaerodynamic structure or a hydrodynamic structure to be optimized usinga parameter set comprising a first number of object parameters; creatingoffsprings of the parameter set by modifying the first number of objectparameters to a second number of object parameters and modifying astructure of the parameter set, the structure of the parameter setdefined by a number and position of the object parameters and strategyparameters; and evaluating quality of the offsprings; wherein theparameter set comprises at least one strategy parameter representing astep-size of the modification of associated object parameters; andstoring one of said optimized model, optimized structure, optimizedshape or optimized design in a computer storage.
 13. The optimizationmethod of claim 12, wherein said step-size of the modification is avariance of a normal distribution.
 14. The optimization method of claim12, wherein said one of the model, structure, shape, or design isdescribed using a spline.
 15. The optimization method of claim 14,wherein the object parameters comprise control points and knot points,the method further comprising adapting a knot vector by inserting newcontrol points and strategy parameters.
 16. The optimization method ofclaim 15, further comprising estimating values of newly insertedstrategy parameters based upon values of strategy parameters ofneighboring control points.
 17. A computer based method for optimizing aspline coded structure representing an aerodynamic structure or ahydrodynamic structure based on an evolution strategy, comprising:describing the spline coded structure to be optimized using a parameterset comprising a first number of object parameters representing controlpoints and knot points and at least one strategy parameter representinga step-size of a mutation of associated object parameters; mutating theobject parameters and strategy parameters to create offsprings of theset, comprising: determining a control point insertion, inserting thecontrol point in the parameter set, inserting an additional strategyparameter for the inserted control point, determining the knot pointsmodified by the insertion of the control point, determining a weightedaveraging of strategy parameter values of modified control points, andassigning the weighted average value as a value of the inserted strategyparameter; evaluating quality of the offsprings; and storing the mutatedobject parameters and strategy parameters in a computer memory.
 18. Themethod of claim 17, wherein said step-size of the mutation is a varianceof a normal distribution.
 19. The method of claim 1, wherein the model,structure, shape or design representing an aerodynamic structure or ahydrodynamic structure comprises one of: an airfoil; a spline codedstructure; a turbine blade for a gas turbine; an aerodynamic structure;or a hydrodynamic structure.
 20. A computer program stored in a computerreadable medium for performing the method of claim
 1. 21. A computerprogram stored in a computer readable medium for performing the methodof claim
 12. 22. A computer program stored in a computer readable mediumfor performing the method of claim 17.